Introduction to Quantum Computation Leila Taghavi August 14, 2021 Postulates of Quantum Mechanics Postulate 1 • Hilbert space: a complex vector space with inner product (state space of the system) • State vector of a qubit: a unit vector in • Computational basis 3 Examples • Superposition: 4 Hilbert space of a composite system Examples: entanglement 5 Postulate 2 • Evolution of a closed system Unitary transformation: Example: 6 Postulate 3 • Measurement in computational basis collapse Examples: collapse collapse 7 Question? • Distinguishing two quantum state collapse 8 Quantum Circuit Model Classical circuits • Universal set of gates: – AND AND OR NOT 0 0 0 0 1 0 1 0 1 – OR 1 0 0 1 0 1 1 1 1 – NOT 10 Classical circuits (cont.) • Proof of universality of AND, OR, NOT by induction: Therefore OR is redundant: 11 Classical circuits - Half adder Half adder 12 Quantum gates • Single-qubit gates Classical NOT 13 Quantum gates (cont.) • More single-qubit gates https://qiskit.org/textbook/ch-states/single-qubit-gates.html 14 Quantum gates (cont.) • Two-qubit gates 15 Quantum gates (cont.) • Another two-qubit gate 16 Quantum gates (cont.) • Three-qubit gate 17 Embedding of classical circuits into quantum ones • Classical gates are not reversible! • Simulating AND using Toffoli: {X, Toffoli} is complete 18 No cloning theorem Fan-out Half adder ≠ 19 Universality of quantum gates • Uncountably many operators (Quantum gates)! • A finite set of gates: approximates any unitary transformation by any arbitrary precision. • {CNOT, Hadamard, phase, Toffoli} 20 Simple Quantum Circuits Bell state generator H Bell basis 22 Bell state generator 23 Quantum Fourier transform (a special case) In general: 24 Teleportation https://qiskit.org/textbook/ch-algorithms/teleportation.html Teleportation • Alice wants to send a qubit to Bob • are two complex numbers! • Can she do it by sending two classical bits?! • Experimentally verified, record distance 143 km 26 Teleportation H send send X Z 27 Superdense Coding https://qiskit.org/textbook/ch-algorithms/superdense-coding.html Superdense coding • Can Alice encode 2 classical bits into a single qubit and send it to Bob? 29 Superdense coding 30 Superdense coding • Sending 2 bits using a single qubit Encoding send H Decoding 31 Deutsch-Jozsa algorithm (1992) https://qiskit.org/textbook/ch-algorithms/deutsch-jozsa.html The problem • Given a function • Decide: is constant or balanced 0 or 1 on all inputs 1 for half, 0 for the other half Classically best case: queries worst case: queries First quantum algorithm that performs better than the best classical algorithm 33 The algorithm 0 or 1 on all inputs 1 for half, 0 for the other half • Is constant or balanced? 1 if constant 0 if balanced Phase kickback 34 Why does it work? Changes in the state of the first qubit: Constant: Balanced: 35 Simon’s Algorithm https://qiskit.org/textbook/ch-algorithms/simon.html The problem • Given a function • Promise: Classically best case: queries worst case: 37 Period finding algorithm of Simon (1994) • • H H H H • 38 About the algorithm • Motivation for QFT based algorithms: Shor’s factoring algorithm • Shor’s algorithm can break RSA 39.